Black hole evaporation 


A brief analysis of the mathematical results of Hawking radiation.  


Black hole evaporation.  
The Hawking temperature T of a Schwarzschild (nonrotating, uncharged) black hole with mass m is given by the equation (in geometrized units) [reference 1]  
T = hbar/(8 pi k m).  equation 1  
In conventional units (which we use here), this would be written  
T = (hbar c^{3})/(8 pi G k m).  equation 2  
The emission of this energy results in an energy decrease of the black
hole, and thus a loss in its mass. What period of time tau will it
take for a black hole of mass mu to evaporate completely? A black hole with mass m has a Schwarzschild radius 

r = 2 G m/c^{2}  equation 3  
and thus an area of  
A = 4 pi r^{2}  equation 4  
A = 16 pi G^{2} m^{2}/c^{4}.  equation 5  
Hawking radiation would have a power P related to the hole's area A and its temperature T by the blackbody power law (with e = 1),  
P = sigma A T^{4}  equation 6  
P = (sigma hbar^{4} c^{8})/(256 pi^{3} G^{2} k^{4} m^{2})  equation 7  
or more conveniently,  
P = K/m^{2}  equation 8  
where K == (sigma hbar^{4} c^{8})/(256 pi^{3} G^{2} k^{4}) = 3.563 x 10^{32} W kg^{2}. Given that the power of the Hawking radiation is the rate of energy loss of the hole, we can write  
P = dE/dt.  equation 9  
Since the total energy E of the hole is related to its mass m by Einstein's massenergy formula,  
E = m c^{2}  equation 10  
we can then rewrite P = dE/dt as  
P = (d/dt) (m c^{2})  equation 11  
P = c^{2} dm/dt.  equation 12  
We can then equate this to our above expression for the power, P = K/m^{2}, and find  
c^{2} dm/dt = K/m^{2}.  equation 13  
This differential equation is separable, and we can write  
m^{2} dm = K/c^{2} dt.  equation 14  
Integrating over m from mu (the initial mass of the hole) to zero (complete evaporation), and over t from zero to tau, we find that  
tau = c^{2}/(3 K) mu^{3}.  equation 15  
That is, the evaporation time of the hole is proportional to the cube
of its mass.


References.  
1. Black holes, white dwarfs, and neutron stars: The physics of compact objects Stuart L. Shapiro, Saul A. Teukolsky p. 366 WileyInterscience; 1983 
reference 1  
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